 Quaternion group

The quaternion group is $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$ with $i^2=j^2=k^2=ijk=-1$. It is a nonabelian group of order eight: $ij=k$ but $ji=-k$.

Its unique element of order two is $-1$, and its other six nonidentity elements have order four. Every subgroup is normal: the three order-four subgroups have index two, and $\{1,-1\}$ is central. Thus it is a nonabelian Dedekind group. It differs from the dihedral group of order eight, which has several elements of order two, despite having the same group order.

The Quaternion group, often denoted as \( Q_8 \), is a specific group in abstract algebra that represents a group of unit quaternions.